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\title{Additional Vector Analysis}

\begin{document}

% \begin{multicols}{2}
    \begin{equation}
        \begin{aligned}
        \mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) &=\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})=\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b}) \\
        \mathbf{a} \times(\mathbf{b} \times \mathbf{c}) &=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \\
        (\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d}) &=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d})-(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) \\
        \nabla \times \nabla \psi &=0 \\
        \nabla \cdot(\nabla \times \mathbf{a}) &=0 \\
        \nabla \times(\nabla \times \mathbf{a}) &=\nabla(\nabla \cdot \mathbf{a})-\nabla^2 \mathbf{a} \\
        \nabla \cdot(\psi \mathbf{a}) &=\mathbf{a} \cdot \nabla \psi+\psi \nabla \cdot \mathbf{a} \\
        \nabla \times(\psi \mathbf{a}) &=\nabla \psi \times \mathbf{a}+\psi \nabla \times \mathbf{a} \\
        \nabla(\mathbf{a} \cdot \mathbf{b}) &=(\mathbf{a} \cdot \nabla) \mathbf{b}+(\mathbf{b} \cdot \nabla) \mathbf{a}+\mathbf{a} \times(\nabla \times \mathbf{b})+\mathbf{b} \times(\nabla \times \mathbf{a}) \\
        \nabla \cdot(\mathbf{a} \times \mathbf{b}) &=\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b}) \\
        \nabla \times(\mathbf{a} \times \mathbf{b}) &=\mathbf{a}(\nabla \cdot \mathbf{b})-\mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}
        \end{aligned}
        \end{equation}

    If $\mathbf x$ is the coordinate of a point with respect to some origin, with magnitude $r=|\mathbf x|$, $\mathbf n=\mathbf x/r$ is a unit radial vector, and $f(r)$ is a well-behaved function of $r$, then

    \begin{equation}
        \begin{aligned}
        &\boldsymbol{\nabla} \cdot \mathbf{x}=3 \quad \nabla \times \mathbf{x}=0\\
        &\boldsymbol{\nabla} \cdot[\mathbf{n} f(r)]=\frac{2}{r} f+\frac{\partial f}{\partial r} \quad \nabla \times[\mathbf{n} f(r)]=0\\
        &(\mathbf{a} \cdot \nabla) \mathbf{n} f(r)=\dfrac{f(r)}{r}[\mathbf{a}-\mathbf{n}(\mathbf{a} \cdot \mathbf{n})]+\mathbf{n}(\mathbf{a} \cdot \mathbf{n}) \frac{\partial f}{\partial r}\\
        &\nabla(\mathbf{x} \cdot \mathbf{a})=\mathbf{a}+\mathbf{x}(\nabla \cdot \mathbf{a})+i(\mathbf{L} \times \mathbf{a})
        \end{aligned}
        \end{equation}
        where $\mathbf L = \dfrac 1i(\mathbf x\times \nabla)$ is the angular-momentum operator.
% \end{multicols}

\end{document}